**MULTIPLICATION OF VECTOR BY A SCALAR**

A vector multiplied by a scalar λ results in another vector, λ. If λ is a
positive number then λ is also in the direction of . If λ
is a negative number, λ is in the opposite direction to the vector .

** **

**Scalar
Product of Two Vectors**

**Definition**

The
scalar product (or dot product) of two vectors is defined as the product of the
magnitudes of both the vectors and the cosine of the angle between them.

Thus
if there are two vectors and having an angle θ
between them, then their scalar product is defined as ⋅ = *AB*
cos θ. Here, *A* and *B* are magnitudes of and .

**Properties**

The
product quantity ⋅ is
always a scalar. It is positive if the angle between the vectors is acute
(i.e., < 90°) and negative if the angle between them is obtuse (i.e.
90°<θ< 180°).

The
work done is basically a scalar product between the force vector and the
displacement vector. Apart from work done, there are other physical quantities
which are also defined through scalar products.

** **

**The
Vector Product of Two Vectors**

**Definition**

*The vector product or cross product
of two vectors is defined as another vector having a magnitude equal to the
product of the magnitudes of two vectors and the sine of the angle between
them. The direction of the product vector is perpendicular to the plane
containing the two vectors, in accordance with the right hand screw rule or
right hand thumb rule *(Figure
2.22)*.*

**Properties of vector (cross) product.**

A
number of quantities used in Physics are defined through vector products.
Particularly physical quantities representing rotational effects like torque,
angular momentum, are defined through vector products.

** **

**Properties
of the components of vectors**

If two vectors and are equal, then their individual components are also equal.

**Solved
Example Problems for Multiplication Of Vector By A Scalar**

**Solved
Example Problems for Scalar Product of Two Vectors**

**Solved
Example Problems for The Vector Product of Two Vectors**

**Solved
Example Problems for Properties of the components of vectors**

Study Material, Lecturing Notes, Assignment, Reference, Wiki description explanation, brief detail

11th Physics : UNIT 2 : Kinematics : Multiplication of Vector by a Scalar | with Solved Example Problems